BIO 377

Life Tables

Today's exercises are intended to familiarize you with life table functions. We will use the program called Populus, which was developed at the University of Minnesota as part of their ecology program. Populus is a set of ecological and population genetic simulations, covering many topics, including life histories, population growth, population interactions, and selection.

Assignment

Start Populus. At the main menu choose "Population Growth". On the population growth menu choose "Age-Structured Population Growth". This will get you into the part of Populus that we will use this week. Read the introductory material to re-familiarize yourself with the concepts associated with age structured populations.

The program for age-structured population growth allows you to set the following parameters: the number of age groups, the l_x and m_x schedules, and the initial number of individuals in each age group. In addition you can choose the number of age classes and the number of time periods for the simulation. To begin, choose 6 age classes (0 to 5), and 20 time periods. You can, of course, change these at any time, should you want to. Remember that by definition l₀ = 1 and m₀ = 0, so don't try to change these values.

Exercise #1 - Annuals

Set up Populus so that l₀ = 1, l₁ = 1, and all other l_x = 0. Set m_x = 0 for all ages except age 1. Note that the l_x used by Populus differs slightly from what you have seen in lecture. Set m₁ to any value that you wish. Set the initial number of individuals to 1 for newborns and zero for all other age groups. Now, let the simulation run by hitting the enter key. Examine the results, return to the parameters table, change the value of m₁, and run the simulation again. Repeat this process and then address the following questions:

1. For annuals, what is the relationship between m_x and the measures of population growth, lambda and r?
2. For annuals, what is the generation time and how does it depend upon m_x?
3. For annuals, what is the pattern of population growth?
   

Exercise #2 - Semelparous (Monocarpic) Perennials

Now allow your annual organism to become a perennial, i.e. allow it to live past age 1. Now vary the age of first and last reproduction: set m_x = 0 for all ages except for one. Set that value of m_x to whatever you like. For the sake of simplicity let all the l_x values be 1 until reproduction and all subsequent values of l_x = 0. Now, do a series of runs in which you increase the age of reproduction from 1 to 4 or more, keeping the number of young produced constant.

1. For semelparous perennials, what is the effect of age of first reproduction on the population growth parameters lambda and r?
2. For semelparous perennials, how does generation time change with age of first reproduction?
3. For semelparous perennials, how does the pattern of population growth vary as age of first reproduction changes?
4. Now, play with both the age of first reproduction and fecundity. For semelparous perennials, how much would you have to increase reproduction in order to compensate for delaying reproduction?
   

Exercises #3 - Overlapping Generations

Now that you are comfortable with the basic model, explore what happens with a more general and more realistic set of parameter values. Allow individuals to breed at more than one age, and allow mortality to occur throughout the life history. For a survivorship schedule, you might try having a constant number or a constant proportion of the population die each time period (e.g. l_x = 1, .8, .6, .4, ... or l_x = 1, .5, .25, .125, ...). Choose a set of reproductive rates in which reproduction occurs in most or all years. Begin by choosing a set of l_x and m_x values that result in a positive growth rate.

1. Keep the total number potential offspring constant, but vary the age of first reproduction. What effect does this have of population growth rates?
2. Keep R₀ constant, but shift the age of first reproduction or shift the weighting of reproduction from early in life to late in life. What effect does this have on population growth rates?
3. For a variety of life histories, examine the pattern of reproductive value. What is the value for newborns? Why isn't the highest value for newborns? At what age does the highest value occur? Why is the highest value at that age?
4. Begin your simulations with just newborns, or just one year olds, or two year olds, etc. Examine the population size at the end of the simulation. Does final population size depend upon the age structure of the initial population? What is the relationship between this pattern and reproductive value?
5. Choose sets of values for which the population exhibits a variety of growth rates from rapid growth to rapid decline. What effect does growth rate have upon the final age structure of the population?
6. Vary the initial number of individuals of different ages for a given set of l_x and m_x values. What effect does the initial age distribution have on the final age distribution? What effect does the initial age distribution have upon the finite rate of population (lambda) during the initial stages of population growth?
7. Set up a life table in which individuals live at least 4 years and breeding occurs only at this maximum age. This, of course, is a semelparous life history. Start with a population composed of only one age group. Do a population projection and then examine how the age structure of the population varies with time. Do this again but start the population with say 2 & 4 year olds, but no 0, 1, or 3 year olds. Project the population and see how the age structure varies. Now allow individuals one year younger than the maximum age to reproduce at a rate about 1/10^th that of the fully mature individuals. Again project the population. What happens here with respect to the age structure over time?